Interpolation of spectra

The interpolation of spectra in SWAN, both in space and time, is a slight modification of the procedure as used in WAM. This procedure is not a simple (spectral) bin-by-bin interpolation because that would cause reduction of the spectral peak if the peaks of the original spectra do not coincide. It is an interpolation where the spectra are first normalized by average frequency and direction, then interpolated and then transformed back.


The average frequencies of the two origin spectra are determined using the frequency moments of the spectra

  $$m_{i,k} = \int N_{i} (\sigma,\theta)\sigma^k d\sigma d\theta$$ (3.128)

with $i$=1,2 (the two origin spectra) and $k$=0,1 (the zero- and first frequency moments of these spectra). Then

  $$\overline{\sigma}_{i} = \frac{m_{i,1}}{m_{i,0}}$$ (3.129)

The average frequency for the interpolated spectrum is calculated as

  $$\overline{\sigma} = (w_{2} m_{1,1} + w_{1} m_{2,1})/(w_{2} m_{1,0} + w_{1} m_{2,0})$$ (3.130)

where $w_{1}$ is the relative distance (in space or time) from the interpolated spectrum to the first origin spectrum $N_{1}(\sigma,\theta)$ and $w_{2}$ is the same for the second origin spectrum $N_{2}(\sigma,\theta)$. Obviously, $w_{1} + w_{2} = 1$.


The average directions of the two origin spectra are determined using directional moments of the spectra:

  $$m_{i,x} = \int N_{i} (\sigma,\theta) \cos (\theta) d\sigma d\theta$$ (3.131)

and

  $$m_{i,y} = \int N_{i} (\sigma,\theta) \sin (\theta) d\sigma d\theta$$ (3.132)

with $i$=1,2. The average direction is then

  $${\overline{\theta}}_{i} = \mbox{atan} (\frac{m_{i,y}}{m_{i,x}})$$ (3.133)

The average direction of the interpolated spectrum is calculated as

  $$\overline{\theta} = \mbox{atan} [\frac{w_{2} m_{1,y} + w_{1} m_{2,y}}{w_{2} m_{1,x} + w_{1} m_{2,x}}]$$ (3.134)

Finally the interpolated spectrum is calculated as follows

  $$N(\sigma,\theta) = w_{2} N_{1} [\overline{\sigma}_{1} \sigma/\ove... ...ma / \overline{\sigma}, \theta - (\overline{\theta}-{\overline{\theta}}_{2})]$$ (3.135)